Title: Special Quadrature Processes for Summation of Slowly Convergent Series

Abstract: Slowly convergent series appear in many problems in mathematics, physics and other sciences. There are several numerical methods based on linear and nonlinear transformations (e.g., Cesàro-transformation, Aitken \(\Delta^2\)-transformation, Wynn-Shanks \(\varepsilon\)-algorithm, \(E\)-algorithm, Levin's transformation, \(\rho\)-algorithm, etc.). In this lecture we present the so-called summation/integration methods, which are very efficient. Methods are based on certain transformations of sums to weighted integrals over the real line or the half-line and on an application of special Gaussian quadrature formulas with respect to some non-classical weight functions (Bose-Einstein, Fermi-Dirac, hyperbolic weights, \(\ldots\)). For constructing such quadrature rules we use a recent progress in symbolic computation and variable-precision arithmetic, implemented through our Mathematica package “OrthogonalPolynomials.” Several interesting applications will also be presented.